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Trigonometry Examples
Step 1
Interchange the variables.
Step 2
Step 2.1
Rewrite the equation as .
Step 2.2
Cross multiply.
Step 2.2.1
Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.
Step 2.2.2
Simplify the left side.
Step 2.2.2.1
Simplify .
Step 2.2.2.1.1
Apply the distributive property.
Step 2.2.2.1.2
Multiply by .
Step 2.3
Rewrite the equation as .
Step 2.4
To remove the radical on the left side of the equation, square both sides of the equation.
Step 2.5
Simplify each side of the equation.
Step 2.5.1
Use to rewrite as .
Step 2.5.2
Simplify the left side.
Step 2.5.2.1
Simplify .
Step 2.5.2.1.1
Apply the product rule to .
Step 2.5.2.1.2
Raise to the power of .
Step 2.5.2.1.3
Multiply by .
Step 2.5.2.1.4
Multiply the exponents in .
Step 2.5.2.1.4.1
Apply the power rule and multiply exponents, .
Step 2.5.2.1.4.2
Cancel the common factor of .
Step 2.5.2.1.4.2.1
Cancel the common factor.
Step 2.5.2.1.4.2.2
Rewrite the expression.
Step 2.5.2.1.5
Simplify.
Step 2.5.3
Simplify the right side.
Step 2.5.3.1
Simplify .
Step 2.5.3.1.1
Rewrite as .
Step 2.5.3.1.2
Expand using the FOIL Method.
Step 2.5.3.1.2.1
Apply the distributive property.
Step 2.5.3.1.2.2
Apply the distributive property.
Step 2.5.3.1.2.3
Apply the distributive property.
Step 2.5.3.1.3
Simplify and combine like terms.
Step 2.5.3.1.3.1
Simplify each term.
Step 2.5.3.1.3.1.1
Multiply by .
Step 2.5.3.1.3.1.2
Multiply by by adding the exponents.
Step 2.5.3.1.3.1.2.1
Move .
Step 2.5.3.1.3.1.2.2
Multiply by .
Step 2.5.3.1.3.1.3
Multiply by by adding the exponents.
Step 2.5.3.1.3.1.3.1
Move .
Step 2.5.3.1.3.1.3.2
Multiply by .
Step 2.5.3.1.3.1.4
Multiply by by adding the exponents.
Step 2.5.3.1.3.1.4.1
Move .
Step 2.5.3.1.3.1.4.2
Multiply by .
Step 2.5.3.1.3.1.5
Multiply .
Step 2.5.3.1.3.1.5.1
Raise to the power of .
Step 2.5.3.1.3.1.5.2
Raise to the power of .
Step 2.5.3.1.3.1.5.3
Use the power rule to combine exponents.
Step 2.5.3.1.3.1.5.4
Add and .
Step 2.5.3.1.3.2
Add and .
Step 2.6
Solve for .
Step 2.6.1
Substitute for .
Step 2.6.2
Since is on the right side of the equation, switch the sides so it is on the left side of the equation.
Step 2.6.3
Add to both sides of the equation.
Step 2.6.4
Subtract from both sides of the equation.
Step 2.6.5
Use the quadratic formula to find the solutions.
Step 2.6.6
Substitute the values , , and into the quadratic formula and solve for .
Step 2.6.7
Simplify the numerator.
Step 2.6.7.1
Apply the distributive property.
Step 2.6.7.2
Multiply by .
Step 2.6.7.3
Multiply by .
Step 2.6.7.4
Rewrite as .
Step 2.6.7.5
Expand using the FOIL Method.
Step 2.6.7.5.1
Apply the distributive property.
Step 2.6.7.5.2
Apply the distributive property.
Step 2.6.7.5.3
Apply the distributive property.
Step 2.6.7.6
Simplify and combine like terms.
Step 2.6.7.6.1
Simplify each term.
Step 2.6.7.6.1.1
Rewrite using the commutative property of multiplication.
Step 2.6.7.6.1.2
Multiply by by adding the exponents.
Step 2.6.7.6.1.2.1
Move .
Step 2.6.7.6.1.2.2
Use the power rule to combine exponents.
Step 2.6.7.6.1.2.3
Add and .
Step 2.6.7.6.1.3
Multiply by .
Step 2.6.7.6.1.4
Multiply by .
Step 2.6.7.6.1.5
Multiply by .
Step 2.6.7.6.1.6
Multiply by .
Step 2.6.7.6.2
Add and .
Step 2.6.7.7
Apply the distributive property.
Step 2.6.7.8
Multiply by by adding the exponents.
Step 2.6.7.8.1
Move .
Step 2.6.7.8.2
Use the power rule to combine exponents.
Step 2.6.7.8.3
Add and .
Step 2.6.7.9
Multiply by .
Step 2.6.7.10
Subtract from .
Step 2.6.7.11
Add and .
Step 2.6.7.12
Add and .
Step 2.6.8
Simplify the expression to solve for the portion of the .
Step 2.6.8.1
Change the to .
Step 2.6.8.2
Factor out of .
Step 2.6.8.3
Rewrite as .
Step 2.6.8.4
Factor out of .
Step 2.6.8.5
Factor out of .
Step 2.6.8.6
Factor out of .
Step 2.6.8.7
Rewrite as .
Step 2.6.8.8
Move the negative in front of the fraction.
Step 2.6.9
Simplify the expression to solve for the portion of the .
Step 2.6.9.1
Simplify the numerator.
Step 2.6.9.1.1
Apply the distributive property.
Step 2.6.9.1.2
Multiply by .
Step 2.6.9.1.3
Multiply by .
Step 2.6.9.1.4
Rewrite as .
Step 2.6.9.1.5
Expand using the FOIL Method.
Step 2.6.9.1.5.1
Apply the distributive property.
Step 2.6.9.1.5.2
Apply the distributive property.
Step 2.6.9.1.5.3
Apply the distributive property.
Step 2.6.9.1.6
Simplify and combine like terms.
Step 2.6.9.1.6.1
Simplify each term.
Step 2.6.9.1.6.1.1
Rewrite using the commutative property of multiplication.
Step 2.6.9.1.6.1.2
Multiply by by adding the exponents.
Step 2.6.9.1.6.1.2.1
Move .
Step 2.6.9.1.6.1.2.2
Use the power rule to combine exponents.
Step 2.6.9.1.6.1.2.3
Add and .
Step 2.6.9.1.6.1.3
Multiply by .
Step 2.6.9.1.6.1.4
Multiply by .
Step 2.6.9.1.6.1.5
Multiply by .
Step 2.6.9.1.6.1.6
Multiply by .
Step 2.6.9.1.6.2
Add and .
Step 2.6.9.1.7
Apply the distributive property.
Step 2.6.9.1.8
Multiply by by adding the exponents.
Step 2.6.9.1.8.1
Move .
Step 2.6.9.1.8.2
Use the power rule to combine exponents.
Step 2.6.9.1.8.3
Add and .
Step 2.6.9.1.9
Multiply by .
Step 2.6.9.1.10
Subtract from .
Step 2.6.9.1.11
Add and .
Step 2.6.9.1.12
Add and .
Step 2.6.9.2
Change the to .
Step 2.6.9.3
Factor out of .
Step 2.6.9.4
Rewrite as .
Step 2.6.9.5
Factor out of .
Step 2.6.9.6
Factor out of .
Step 2.6.9.7
Factor out of .
Step 2.6.9.8
Rewrite as .
Step 2.6.9.9
Move the negative in front of the fraction.
Step 2.6.10
The final answer is the combination of both solutions.
Step 2.6.11
Substitute for .
Step 2.6.12
Set up each of the solutions to solve for .
Step 2.6.13
Solve for in .
Step 2.6.13.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 2.6.13.2
Simplify the right side.
Step 2.6.13.2.1
Simplify .
Step 2.6.13.2.1.1
Split the fraction into two fractions.
Step 2.6.13.2.1.2
Simplify each term.
Step 2.6.13.2.1.2.1
Split the fraction into two fractions.
Step 2.6.13.2.1.2.2
Simplify each term.
Step 2.6.13.2.1.2.2.1
Cancel the common factor of .
Step 2.6.13.2.1.2.2.1.1
Cancel the common factor.
Step 2.6.13.2.1.2.2.1.2
Rewrite the expression.
Step 2.6.13.2.1.2.2.2
Cancel the common factor of .
Step 2.6.13.2.1.2.2.2.1
Cancel the common factor.
Step 2.6.13.2.1.2.2.2.2
Rewrite the expression.
Step 2.6.13.2.1.2.3
Move the negative in front of the fraction.
Step 2.6.13.2.1.3
Apply the distributive property.
Step 2.6.13.2.1.4
Simplify.
Step 2.6.13.2.1.4.1
Multiply by .
Step 2.6.13.2.1.4.2
Multiply .
Step 2.6.13.2.1.4.2.1
Multiply by .
Step 2.6.13.2.1.4.2.2
Multiply by .
Step 2.6.13.3
Divide each term in by and simplify.
Step 2.6.13.3.1
Divide each term in by .
Step 2.6.13.3.2
Simplify the left side.
Step 2.6.13.3.2.1
Cancel the common factor of .
Step 2.6.13.3.2.1.1
Cancel the common factor.
Step 2.6.13.3.2.1.2
Divide by .
Step 2.6.14
Solve for in .
Step 2.6.14.1
Take the inverse cosine of both sides of the equation to extract from inside the cosine.
Step 2.6.14.2
Simplify the right side.
Step 2.6.14.2.1
Simplify .
Step 2.6.14.2.1.1
Split the fraction into two fractions.
Step 2.6.14.2.1.2
Simplify each term.
Step 2.6.14.2.1.2.1
Split the fraction into two fractions.
Step 2.6.14.2.1.2.2
Simplify each term.
Step 2.6.14.2.1.2.2.1
Cancel the common factor of .
Step 2.6.14.2.1.2.2.1.1
Cancel the common factor.
Step 2.6.14.2.1.2.2.1.2
Rewrite the expression.
Step 2.6.14.2.1.2.2.2
Cancel the common factor of .
Step 2.6.14.2.1.2.2.2.1
Cancel the common factor.
Step 2.6.14.2.1.2.2.2.2
Rewrite the expression.
Step 2.6.14.2.1.3
Simplify by multiplying through.
Step 2.6.14.2.1.3.1
Apply the distributive property.
Step 2.6.14.2.1.3.2
Multiply by .
Step 2.6.14.3
Divide each term in by and simplify.
Step 2.6.14.3.1
Divide each term in by .
Step 2.6.14.3.2
Simplify the left side.
Step 2.6.14.3.2.1
Cancel the common factor of .
Step 2.6.14.3.2.1.1
Cancel the common factor.
Step 2.6.14.3.2.1.2
Divide by .
Step 2.6.15
List all of the solutions.
Step 3
Replace with to show the final answer.
Step 4
Step 4.1
The domain of the inverse is the range of the original function and vice versa. Find the domain and the range of and and compare them.
Step 4.2
Find the domain of .
Step 4.2.1
Set the radicand in greater than or equal to to find where the expression is defined.
Step 4.2.2
Solve for .
Step 4.2.2.1
Subtract from both sides of the inequality.
Step 4.2.2.2
Divide each term in by and simplify.
Step 4.2.2.2.1
Divide each term in by .
Step 4.2.2.2.2
Simplify the left side.
Step 4.2.2.2.2.1
Cancel the common factor of .
Step 4.2.2.2.2.1.1
Cancel the common factor.
Step 4.2.2.2.2.1.2
Divide by .
Step 4.2.2.2.3
Simplify the right side.
Step 4.2.2.2.3.1
Move the negative in front of the fraction.
Step 4.2.2.3
Since the left side has an even power, it is always positive for all real numbers.
All real numbers
All real numbers
Step 4.2.3
Set the argument in greater than or equal to to find where the expression is defined.
Step 4.2.4
Solve for .
Step 4.2.4.1
Solve for .
Step 4.2.4.1.1
Move all terms not containing to the right side of the inequality.
Step 4.2.4.1.1.1
Add to both sides of the inequality.
Step 4.2.4.1.1.2
Add to both sides of the inequality.
Step 4.2.4.1.1.3
Add and .
Step 4.2.4.1.1.4
Add and .
Step 4.2.4.1.2
Since the expression on each side of the equation has the same denominator, the numerators must be equal.
Step 4.2.4.2
To remove the radical on the left side of the inequality, square both sides of the inequality.
Step 4.2.4.3
Simplify each side of the inequality.
Step 4.2.4.3.1
Use to rewrite as .
Step 4.2.4.3.2
Simplify the left side.
Step 4.2.4.3.2.1
Simplify .
Step 4.2.4.3.2.1.1
Multiply the exponents in .
Step 4.2.4.3.2.1.1.1
Apply the power rule and multiply exponents, .
Step 4.2.4.3.2.1.1.2
Cancel the common factor of .
Step 4.2.4.3.2.1.1.2.1
Cancel the common factor.
Step 4.2.4.3.2.1.1.2.2
Rewrite the expression.
Step 4.2.4.3.2.1.2
Simplify.
Step 4.2.4.3.3
Simplify the right side.
Step 4.2.4.3.3.1
One to any power is one.
Step 4.2.4.4
Solve for .
Step 4.2.4.4.1
Move all terms not containing to the right side of the inequality.
Step 4.2.4.4.1.1
Subtract from both sides of the inequality.
Step 4.2.4.4.1.2
Subtract from .
Step 4.2.4.4.2
Divide each term in by and simplify.
Step 4.2.4.4.2.1
Divide each term in by .
Step 4.2.4.4.2.2
Simplify the left side.
Step 4.2.4.4.2.2.1
Cancel the common factor of .
Step 4.2.4.4.2.2.1.1
Cancel the common factor.
Step 4.2.4.4.2.2.1.2
Divide by .
Step 4.2.4.4.2.3
Simplify the right side.
Step 4.2.4.4.2.3.1
Divide by .
Step 4.2.4.4.3
Since the left side has an even power, it is always positive for all real numbers.
All real numbers
All real numbers
All real numbers
Step 4.2.5
Set the argument in less than or equal to to find where the expression is defined.
Step 4.2.6
Solve for .
Step 4.2.6.1
Solve for .
Step 4.2.6.1.1
Move all terms not containing to the right side of the inequality.
Step 4.2.6.1.1.1
Add to both sides of the inequality.
Step 4.2.6.1.1.2
Add to both sides of the inequality.
Step 4.2.6.1.1.3
Add and .
Step 4.2.6.1.2
Multiply both sides by .
Step 4.2.6.1.3
Simplify.
Step 4.2.6.1.3.1
Simplify the left side.
Step 4.2.6.1.3.1.1
Simplify .
Step 4.2.6.1.3.1.1.1
Rewrite using the commutative property of multiplication.
Step 4.2.6.1.3.1.1.2
Cancel the common factor of .
Step 4.2.6.1.3.1.1.2.1
Factor out of .
Step 4.2.6.1.3.1.1.2.2
Cancel the common factor.
Step 4.2.6.1.3.1.1.2.3
Rewrite the expression.
Step 4.2.6.1.3.1.1.3
Cancel the common factor of .
Step 4.2.6.1.3.1.1.3.1
Cancel the common factor.
Step 4.2.6.1.3.1.1.3.2
Rewrite the expression.
Step 4.2.6.1.3.2
Simplify the right side.
Step 4.2.6.1.3.2.1
Simplify .
Step 4.2.6.1.3.2.1.1
Simplify by multiplying through.
Step 4.2.6.1.3.2.1.1.1
Apply the distributive property.
Step 4.2.6.1.3.2.1.1.2
Simplify the expression.
Step 4.2.6.1.3.2.1.1.2.1
Multiply by .
Step 4.2.6.1.3.2.1.1.2.2
Rewrite using the commutative property of multiplication.
Step 4.2.6.1.3.2.1.2
Simplify each term.
Step 4.2.6.1.3.2.1.2.1
Cancel the common factor of .
Step 4.2.6.1.3.2.1.2.1.1
Factor out of .
Step 4.2.6.1.3.2.1.2.1.2
Cancel the common factor.
Step 4.2.6.1.3.2.1.2.1.3
Rewrite the expression.
Step 4.2.6.1.3.2.1.2.2
Cancel the common factor of .
Step 4.2.6.1.3.2.1.2.2.1
Cancel the common factor.
Step 4.2.6.1.3.2.1.2.2.2
Rewrite the expression.
Step 4.2.6.2
To remove the radical on the left side of the inequality, square both sides of the inequality.
Step 4.2.6.3
Simplify each side of the inequality.
Step 4.2.6.3.1
Use to rewrite as .
Step 4.2.6.3.2
Simplify the left side.
Step 4.2.6.3.2.1
Simplify .
Step 4.2.6.3.2.1.1
Multiply the exponents in .
Step 4.2.6.3.2.1.1.1
Apply the power rule and multiply exponents, .
Step 4.2.6.3.2.1.1.2
Cancel the common factor of .
Step 4.2.6.3.2.1.1.2.1
Cancel the common factor.
Step 4.2.6.3.2.1.1.2.2
Rewrite the expression.
Step 4.2.6.3.2.1.2
Simplify.
Step 4.2.6.3.3
Simplify the right side.
Step 4.2.6.3.3.1
Simplify .
Step 4.2.6.3.3.1.1
Rewrite as .
Step 4.2.6.3.3.1.2
Expand using the FOIL Method.
Step 4.2.6.3.3.1.2.1
Apply the distributive property.
Step 4.2.6.3.3.1.2.2
Apply the distributive property.
Step 4.2.6.3.3.1.2.3
Apply the distributive property.
Step 4.2.6.3.3.1.3
Simplify and combine like terms.
Step 4.2.6.3.3.1.3.1
Simplify each term.
Step 4.2.6.3.3.1.3.1.1
Rewrite using the commutative property of multiplication.
Step 4.2.6.3.3.1.3.1.2
Multiply by by adding the exponents.
Step 4.2.6.3.3.1.3.1.2.1
Move .
Step 4.2.6.3.3.1.3.1.2.2
Use the power rule to combine exponents.
Step 4.2.6.3.3.1.3.1.2.3
Add and .
Step 4.2.6.3.3.1.3.1.3
Multiply by .
Step 4.2.6.3.3.1.3.1.4
Multiply by .
Step 4.2.6.3.3.1.3.1.5
Multiply by .
Step 4.2.6.3.3.1.3.1.6
Multiply by .
Step 4.2.6.3.3.1.3.2
Add and .
Step 4.2.6.4
Solve for .
Step 4.2.6.4.1
Rewrite so is on the left side of the inequality.
Step 4.2.6.4.2
Move all terms containing to the left side of the inequality.
Step 4.2.6.4.2.1
Subtract from both sides of the inequality.
Step 4.2.6.4.2.2
Combine the opposite terms in .
Step 4.2.6.4.2.2.1
Subtract from .
Step 4.2.6.4.2.2.2
Add and .
Step 4.2.6.4.3
Move all terms not containing to the right side of the inequality.
Step 4.2.6.4.3.1
Subtract from both sides of the inequality.
Step 4.2.6.4.3.2
Subtract from .
Step 4.2.6.4.4
Divide each term in by and simplify.
Step 4.2.6.4.4.1
Divide each term in by .
Step 4.2.6.4.4.2
Simplify the left side.
Step 4.2.6.4.4.2.1
Cancel the common factor of .
Step 4.2.6.4.4.2.1.1
Cancel the common factor.
Step 4.2.6.4.4.2.1.2
Divide by .
Step 4.2.6.4.4.3
Simplify the right side.
Step 4.2.6.4.4.3.1
Divide by .
Step 4.2.6.4.5
Since the left side has an even power, it is always positive for all real numbers.
All real numbers
All real numbers
All real numbers
Step 4.2.7
Set the denominator in equal to to find where the expression is undefined.
Step 4.2.8
Solve for .
Step 4.2.8.1
Divide each term in by and simplify.
Step 4.2.8.1.1
Divide each term in by .
Step 4.2.8.1.2
Simplify the left side.
Step 4.2.8.1.2.1
Cancel the common factor of .
Step 4.2.8.1.2.1.1
Cancel the common factor.
Step 4.2.8.1.2.1.2
Divide by .
Step 4.2.8.1.3
Simplify the right side.
Step 4.2.8.1.3.1
Divide by .
Step 4.2.8.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 4.2.8.3
Simplify .
Step 4.2.8.3.1
Rewrite as .
Step 4.2.8.3.2
Pull terms out from under the radical, assuming positive real numbers.
Step 4.2.8.3.3
Plus or minus is .
Step 4.2.9
The domain is all values of that make the expression defined.
Step 4.3
Since the domain of is not equal to the range of , then is not an inverse of .
There is no inverse
There is no inverse
Step 5